Line graphs and 22-geodesic transitivity

For a graph $\Gamma$, a positive integer $s$ and a subgroup G\leq \Aut(\Gamma), we prove that $G$ is transitive on the set of $s$-arcs of $\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive

Symmetry properties of subdivision graphs

The subdivision graph $S(\Sigma)$ of a graph $\Sigma$ is obtained from $\Sigma$ by `adding a vertex' in the middle of every edge of \Si. Various symmetry properties of $\S(\Sigma)$ are studied. We prove that, for a connected graph $\Sigma$, $S(\Sigma)$ is locally $s$-arc transitive if and only if $\Sigma$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. The diameter of $S(\Sigma)$ is $2d+\delta$, where $\Sigma$ has diameter $d$ and $0\leqslant \delta\leqslant 2$, and local $s$-distance transitivity of $\S(\Sigma)$ is defined for $1\leqslant s\leqslant 2d+\delta$. In the general case where $s\leqslant 2d-1$ we prove that $S(\Sigma)$ is locally $s$-distance transitive if and only if $\Sigma$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. For the remaining values of $s$, namely $2d\leqslant s\leqslant 2d+\delta$, we classify the graphs $\Sigma$ for which $S(\Sigma)$ is locally $s$-distance transitive in the cases, $s\leqslant 5$ and $s\geqslant 15+\delta$. The cases $\max\{2d, 6\}\leqslant s\leqslant \min\{2d+\delta, 14+\delta\}$ remain open